What will be the estimator for these parameters

Question

Question:

$y_0 = z^d$ is computed from the sum of some recordings by a sensor. Let, there be $k$ sensor nodes. This parameter is calculated by each sensor node and then transmitted to the base station which is the receiver where the channel noise is an Additive White Gaussian noise. So, at each $t$ time instant, a new value of this parameter is received, but is a noisy version of the true parameter.

$y(t) = y_0 + \eta(t)$ for $t=1,2,..,T$ are the observations

$d<T$ and is a positive known integer.

$\eta(t) = N(0,\sigma_\eta^2)$

The distribution of $z^d$ is generalized Gamma (How to compute the pdf analytically for the sum of generalized gamma).

Statistical Information about the parameter:

$$z^d = \sum_{i=1}^k X_i^d$$ and $$\sum_{i=1}^k X_i^d\sim\text{G}(kn,\lambda)$$. $Z$ has a Generalized Gamma distribution and

$z^d = \sum_i^{k} X_i^d = G(kn, \lambda)$

Assuming, $\lambda =1$ and $k,n,d$ are known.

I need to find an estimator for the parameters $y_0, \sigma_\eta^2)$ from the observations $y(t)$,in order to get an estimator for the parameter $z^d$.

Assuming, $\lambda =1$.

This is how I proceeded.

Formulating a Maximum likelihood estimator:

So, the likelihood will be

$p(y;\mathbf{h}) = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y - y_0(t))^2)/ 2\sigma_\eta^2}$. $ = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y(t) - z^d)^2)/ 2\sigma_\eta^2}$

$ = \frac{1}{{(2 \pi \sigma_\eta^2)}^{T/2}} \exp{(-(y(t) - \sum_i x_i^d)^2)/ 2\sigma_\eta^2}$

Then, I need to differentiate w.r.t the unknowns and equate to zero.

The MLE of $x$ is the value that minimizes the likelihood function or equivalently the value that minimized

$J(x) = \sum_{t=0}^{T-1} {((y(t) - z^d)}^2$

Differentiating $J(x)$ and setting it to to zero produces

$\sum_{t=0}^{T-1} ((y(t) - z^d) dz^{d-1} = 0$

This is a nonlinear equation in $x$ and cannot be solved directly. Newton-Raphson is a method but it works only for very close initial guess to $x$ and $d$. So, Was thinking how to apply Expectation - Maximization.

I don't know if my approach is correct or not. After this I am stuck because the answer is zero. So, how can I apply expectation maximization

Answer 1

I need to find an estimator for the parameters $(y_0,σ^2_η)$ from the observations $y(t)$,in order to get an estimator for the parameter $z_d$.

In your formulation, $$y(t) = y_0 + \eta(t)$$with $\eta(t)\stackrel{\text{iid}}{\sim}N(0,\sigma^2_\eta)$, i.e.$$y(t) \stackrel{\text{iid}}{\sim}N(y_0,\sigma^2_\eta)\qquad t=1,\ldots,T$$Therefore, if you are only interested in estimating $(y_0,σ^2_η)$, this is a standard normal model with MLEs$$\hat y_0=\frac{1}{T}\sum_{t=1}^T y(t)\quad \hat{σ^2_η}=\frac{1}{T}\sum_{t=1}^T (y(t)-\hat y_0)^2$$ If you assume further that$$y_0=z^d$$then$$\hat z=\hat{y_0}^{1/d}$$since the MLE of the transform is the transform of the MLE (assuming $\hat y_0>0$).

As I mentioned in a comment, the information that $z$ is a generalised Gamma variate is not useful to estimate $z$.

Update

Now, if repeated observations are available, with $y_0(t)=z(t)^d$, and $y_0(t)\sim \text{Ga}(kn,\lambda)$, this item of information does not contain anything about $d$. Once $y_0$ is observed or estimated, you cannot derive an estimator of $d$ because the assumption for the Gamma distribution is on $y_0(t)$, not $z(t)$. The parameter $d$ is not identifiable for this experiment.